Finite Group Databasebeta

← #498625

Subgroups — #498625

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33706C211yesmaximal, characteristic
2#23189C211yesmaximal, characteristic
3#23171C211yesmaximal, characteristic, complemented
4#13278C211yesmaximal, characteristic, complemented
5#13258C211yesmaximal, characteristic
6#13155C211yesmaximal, characteristic
7#13154C211yesmaximal, characteristic, complemented
8#13142C211yesmaximal, characteristic
9#13141C211yesmaximal, characteristic, complemented
10#12957C211yesmaximal, characteristic, complemented
11#12956C211yesmaximal, characteristic
12C4×#365C211yesmaximal, characteristic, complemented
13#12666C211yesmaximal, characteristic, complemented
14#12648C211yesmaximal, characteristic, complemented
15#12647C211yesmaximal, characteristic
16#3160C2211yescharacteristic
17C2×#495C2211yescharacteristic
18#2962—21no—
19C2×#442C2211yescharacteristic
20C2×#441C2211yescharacteristic
21#2367—21no—
22#2243C412yescomplemented
23#2243C412yescomplemented
24#2242C412yescomplemented
25#2242C412yescomplemented
26C2×#365C2211yescharacteristic
27C2×#365C2211yescharacteristic
28#2150C2211yescharacteristic
29C2×#362C2211yescharacteristic
30C2×#362C2211yescharacteristic
31C2×#361C2211yescharacteristic
32C2×#361C2211yescharacteristic
33#2141C2211yescharacteristic
34#2141C2211yescharacteristic
35#2140C2211yescharacteristic
36#2140C2211yescharacteristic
37C2×#347C2211yescharacteristic
38C2×#347C2211yescharacteristic
39C4×M32C2211yescharacteristic
40#1986—21no—
41#1983—21no—
42#1972C2211yescharacteristic
43#1971C2211yescharacteristic
44#1946C2211yescharacteristic
45#1946C2211yescharacteristic
46#1922—21no—
47#1922—21no—
48#1880—21no—
49#1877—21no—
50C4×#108C2211yescharacteristic
51C4×#106C2211yescharacteristic
52#1773C2211yescharacteristic
53#1773C2211yescharacteristic
54#1770C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1386C2211yescharacteristic
58#1386C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×SD16—21no—
65C22×D8C2311yescharacteristic
66#533—21no—
67C2×#124—21no—
68C2×C4×Q8—21no—
69C2×C4×D4—21no—
70C2×M32C2311yescharacteristic
71C2×M32C2311yescharacteristic
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74#500C2×C411yescharacteristic, complemented
75#500C2×C411yescharacteristic, complemented
76#499—21no—
77#499C2×C411yescharacteristic, complemented
78#499C2×C411yescharacteristic, complemented
79#498—21no—
80#495—21no—
81#495—21no—
82#495C2×C412yes—
83#495C2×C412yes—
84#446—41no—
85#446—41no—
86#446—41no—
87C8×Q8—41no—
88#442—22no—
89#442—41no—
90#441—22no—
91#441—41no—
92C2×#108C2311yescharacteristic
93C2×#108C2311yescharacteristic
94#427—21no—
95#427C2311yescharacteristic
96#427C2311yescharacteristic
97C2×#106C2311yescharacteristic
98C2×#106C2311yescharacteristic
99#424—21no—
100#424—21no—
101C2×C4⋊C8D411yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106#405—21no—
107#405—21no—
108C4×M16—21no—
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#390—21no—
114#388—41no—
115#388—41no—
116#385—41no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D411yescharacteristic
119C4×C22⋊C4—41no—
120#365—22no—
121#365—22no—
122C4⋊C16—21no—
123C4⋊C16—21no—
124#362—22no—
125#362—41no—
126#361—22no—
127#361—41no—
128#358C2×C412yescomplemented
129#358C2×C412yescomplemented
130#358C2×C412yescomplemented
131#358C2×C412yescomplemented
132#357C2×C412yes—
133#357C2×C412yes—
134#357C2×C412yes—
135#357C2×C412yes—
136#347—41no—
137#347—41no—
138#346C2×C412yescomplemented
139C4×C16C2×C412yescomplemented
140#341—21no—
141#340—21no—
142#339—21no—
143#339—21no—
144#339—21no—
145#339—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148C22×Q8—21no—
149C22×D4—21no—
150C23×C4—41no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16C22×C412yes—
157C2×SD16—22no—
158C2×SD16—22no—
159C2×SD16—81no—
160C2×SD16—81no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8C22×C412yes—
167C2×M16—21no—
168C2×M16—21no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2411yescharacteristic
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175#125—21no—
176#125—21no—
177#124—22no—
178#124—22no—
179#124—41no—
180#124—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C4×Q8—42no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—81no—
187C4×D4—42no—
188C4×D4—81no—
189#117—21no—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205M32—21no—
206M32—21no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16C22×C412yes—
216C2×C16C22×C412yes—
217#108—21no—
218#108—21no—
219#108—21no—
220#108#4112yes—
221#107—21no—
222#107—21no—
223#107C22×C411yescharacteristic
224#107C22×C411yescharacteristic
225#107C22×C411yescharacteristic
226#107C22×C411yescharacteristic
227#106—21no—
228#106—21no—
229#106—21no—
230#106—21no—
231#106—21no—
232#106—21no—
233#106—21no—
234#106#4112yes—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241#103—42no—
242#103—81no—
243#102—42no—
244#102—81no—
245#97—21no—
246#97—21no—
247#97—21no—
248#97—21no—
249#97—41no—
250#97—41no—
251#97—41no—
252C4×C8—41no—
253C4×C8—41no—
254C4×C8—41no—
255C4×C8C22×C411yescharacteristic
256C4×C8C22×C411yescharacteristic
257C4×C8#4111yescharacteristic
258C4×C8#4111yescharacteristic
259#95—21no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24—41no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—161no—
286C22×C4AGL(1,ℤ/8)11yescharacteristic
287C22×C4AGL(1,ℤ/8)11yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—41no—
292Q16—41no—
293SD16—82no—
294SD16—82no—
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299M16—22no—
300M16—41no—
301M16—41no—
302M16—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8C4×D411yescharacteristic
314C2×C8C4×D411yescharacteristic
315C2×C8C23×C411yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C82−1+411yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—81no—
336C4⋊C4—82no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—41no—
351C42—41no—
352C42—82no—
353C42—161no—
354C42C4×D411yescharacteristic
355C42C4×D411yescharacteristic
356C42AGL(1,ℤ/8)11yescharacteristic
357C42AGL(1,ℤ/8)11yescharacteristic
358C16—22no—
359C16—22no—
360C16—41no—
361C16—41no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23C2×AGL(1,ℤ/8)11yescharacteristic
370Q8—42no—
371Q8—42no—
372Q8—82no—
373Q8—81no—
374Q8—81no—
375Q8—161no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—81no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—164no—
402C2×C4#44211yescharacteristic
403C2×C4#44211yescharacteristic
404C2×C4#44211yescharacteristic
405C2×C4#44211yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4#54611yescharacteristic
408C2×C4#54811yescharacteristic
409C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57711yescharacteristic
412C8—21no—
413C8—21no—
414C8—41no—
415C8—41no—
416C8—42no—
417C8—81no—
418C8C4×#4111yescharacteristic
419C8#52011yescharacteristic
420C8#55611yescharacteristic
421C8#55911yescharacteristic
422C22—21no—
423C22—21no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22C2×#44211yescharacteristic
432C22#332411yescharacteristic
433C22#336811yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—81no—
439C4—81no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—161no—
447C4#236711yescharacteristic
448C4#294911yescharacteristic
449C4#333211yescharacteristic
450C4#337311yescharacteristic
451C2—21no—
452C2—21no—
453C2—162no—
454C2#2322011yescharacteristic
455C2#3346811yescharacteristic
456C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #23171C2semidirect
4. #13278C2semidirect
7. #13154C2semidirect
9. #13141C2semidirect
10. #12957C2semidirect
12. C4×#365C2semidirect
13. #12666C2semidirect
14. #12648C2semidirect
22. #2243C4semidirect
23. #2243C4semidirect
24. #2242C4semidirect
25. #2242C4semidirect
74. #500C2×C4semidirect
75. #500C2×C4semidirect
77. #499C2×C4semidirect
78. #499C2×C4semidirect
128. #358C2×C4semidirect
129. #358C2×C4semidirect
130. #358C2×C4semidirect
131. #358C2×C4semidirect
138. #346C2×C4semidirect
139. C4×C16C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#337062
#231892
#231712
#132782
#132582
#131552
#131542
#131422
#131412
#129572
#129562
C4×#3652
#126662
#126482
#126472